MLR models have been used to analyse drivers of secular trends and variability in stratospheric ozone for many decades (see e.g. , and references therein). In the context of extracting the SOR from ozone time series, there is a long history of similar methods being applied to both satellite observations (e.g., ) and CCMs . Here we follow the methodology described by , which is very similar to the methods described in those earlier studies. Briefly, the zonal mean ozone data are deseasonalized by removing the long-term monthly mean at each latitude and pressure level. As in past studies, we then perform an MLR analysis on the time series of monthly mean anomalies at each location, O3′(t), to diagnose the solar cycle component: O3′(t)=A×F10.7(t)+B×CO2(t)+C×EESC(t)+D×ENSO(t)+E×QBOA(t)+F×QBOB(t)+r(t), where r(t) is a residual. The annual-mean SOR is calculated by regressing all months as a single time series. The monthly SOR is calculated by regressing time series of year-to-year anomalies for individual months. The monthly mean basis functions in Eq. () are the F10.7cm radio solar flux, the CO2 concentration at Mauna Loa, the EESC and the Niño 3.4 index to represent ENSO. The F10.7cm flux is used to represent solar activity because it has been shown to be well correlated with indices for UV radiation (e.g. ), the key driver of the stratospheric ozone response. The results presented in Sect.  assume a difference of 130 solar flux units (1 SFU = 10-22 Wm-2Hz-1) as a representative amplitude of the 11-year solar cycle. The Niño 3.4 index is computed as the standardized mean SST averaged over the region 5∘ S–5∘ N, 120–170∘ W. For those CCMI-1 models and ozone databases that include QBO variability (see Table 1), the QBO indices are calculated as the first two principal component time series of the tropical (±10∘, 5–70 hPa) zonal mean zonal winds. The ozone response to volcanic aerosols is non-linear through time owing to changing levels of inorganic chlorine in the atmosphere . Thus, rather than including a term in the MLR model to represent volcanic effects on ozone, data from the 2-year periods following the three major tropical volcanic eruptions since 1960 are excluded from the analysis: Mount Agung (February 1963), El Chićhon (March 1982) and Mount Pinatubo (June 1991). Figure  shows example time series of the MLR basis functions from 1960 to 2009 in arbitrary units. In this example the ENSO and QBO indices are based on observations. The coefficients A–F in Eq. () are calculated using linear least squares regression.

One important issue for MLR analysis is the handling of possible autocorrelation in the regression residuals, r(t), and the effect on the estimation of statistical uncertainty in the results. A Durbin–Watson test reveals significant serial correlation in the regression residuals in many locations for lags of 1 and 2 months, particularly in the middle and polar lower stratosphere. Such serial correlation can lead to spurious overestimation of the statistical significance of the regression coefficients, and we therefore include an autoregressive term in the regression model. Given the significant serial correlations in some regions up to a lag of 2 months, a second-order autoregressive noise process (AR2) is used, which assumes the residuals r(t) have the form r(t)=ar(t-1)+br(t-2)+w(t), where a and b are constants and w(t) is a white noise process. This is identical to the approach employed in and the recent SPARC SI2N analysis of ozone trends . The autocorrelation term is not included in the analysis of the monthly SOR because the residuals are approximately uncorrelated from year to year. Unless otherwise stated, the statistical significance of the SOR extracted using the MLR model is assessed using a two-tailed Student's t test with a null hypothesis that the magnitude of the SOR is indistinguishable from zero. We apply a threshold to determine whether the null hypothesis can be rejected at a 95 % confidence level.

It is a challenge in geophysical science to develop statistical methods to extract forced signals from complex time series. The implementation of MLR analysis as described above may have a number of limitations, including (but not limited to) assumption that the input basis functions have zero uncertainty, difficulties in separating a signal from noise in relatively short or sparse records and issues arising from degeneracy between basis functions . We have not attempted to account for these factors in our results.

To explore the atmospheric impacts of different representations of the SOR, simulations were carried out with the atmospheric general circulation model ECHAM6.3, which is an update of the ECHAM6.1 model used as the atmospheric component of the Max Planck Institute Earth System Model in CMIP5 simulations. Model changes from version 6.1 to 6.3 are mainly related to fixes of bugs described by ; efforts to ensure energy conservations; an update of the radiation scheme, which is now the PSrad version of the RRTMG code ; and retuning. If the same forcings are used, temperature effects of solar cycle variability in ECHAM6.3 compare well to those described for ECHAM6.1 . The model experiments performed here use a horizontal resolution of T63 (∼ 140 km × 210 km) with 47 vertical levels up to a lid of 80 km.

It is known that the ECHAM6.3 radiation code does not cover wavelengths below 200 nm, and therefore the important Schumann–Runge bands and Lyman-alpha lines of ozone are not captured . This results in a too-weak radiative response to the imposed solar forcing particularly in the mesosphere. Therefore we focus our analysis on the stratospheric response, where most of the absorption occurs at higher UV wavelengths, and the performance of ECHAM6.3 is comparable to models with a more comprehensive radiative code .

We have performed five time-slice simulations with ECHAM6.3, each lasting for 50 years. The control simulation uses average boundary conditions as specified for the CMIP5 AMIP simulation; i.e. for all boundary conditions such as SSTs, greenhouse gas concentrations, SSI and prescribed atmospheric ozone we have used multi-year averages of the CMIP5 recommended values for the years 1978–2008. Four sensitivity simulations have then been performed in which solar maximum minus solar minimum differences in either atmospheric ozone concentrations or both ozone and SSI have been added to the respective fields of the control simulation. For solar maximum and minimum conditions we have used average values over the years 1985–1986 and 1981–1982, respectively. According to the solar irradiance recommendations for CMIP6 these are characterized by a difference of 126.1 SFU and are therefore closely comparable to the results presented for the SOR, which assume a representative solar cycle amplitude of 130 SFU. Ozone anomalies were calculated either from the respective years of the Extended CMIP5 ozone database () or using the monthly SOR coefficients from the CMIP6 ozone database shown in Sect. . The corresponding SSI anomalies are calculated either from the CMIP5 recommended NRLSSI-1 dataset or from the CMIP6 recommended solar forcing dataset .

Figure  shows time series of deseasonalized tropical (30∘ S–30∘ N) and monthly mean percent ozone anomalies at select pressure levels (1, 3, 5, 10, 30 hPa) for the eight CCMI-1 models described in Sect. . Anomalies are defined relative to the period 1985–2009. Also plotted in Fig.  are time series from two satellite datasets discussed in Part 1 of this study: SBUVMOD VN8.6 (black dashed) and SAGE-GOMOS 1 (black solid). For completeness, the time series of absolute ozone mixing ratios from the models are shown in the Supplement (Fig. S1).

The CCMI-1 models show a long-term decline in stratospheric ozone abundances, particularly in the mid- and upper stratosphere. This is the result of increasing atmospheric loading of inorganic chlorine and bromine over this period and is consistent with results from earlier CCM studies (e.g. ). At 1 hPa, the trend in ozone between 1979 and 1997 computed by linear regression ranges from -1.9 to -2.6 %decade-1 across the models. At 3 hPa, the range in trends is -4.1 to -5.1 %decade-1. These values are within the uncertainty bounds of satellite-observed ozone trends over this period .

In addition to a long-term decline, Fig.  shows quasi-decadal variations in ozone in the upper stratosphere that are approximately in phase with the 11-year solar cycle. These are a marker of the SOR which is evident in the raw ozone time series and can be seen as a peak around the decadal timescale in the modelled ozone power spectra (see Fig. S2). There is larger interannual and multi-year variability in ozone at 10 and 30 hPa, where some models show enhanced variability associated with the QBO. The modelled evolution of the tropical ozone anomalies is generally in good agreement with the observation data in Fig. , with some exceptions where the satellite records show larger-amplitude short-term fluctuations that may be related to incomplete spatial and temporal sampling.

Figure  shows latitude–pressure cross sections of the annual mean SOR in the eight CCMI-1 models (Fig. a–h) along with the multi-model mean (Fig. i). For the individual models, the statistical significance of the SOR is assessed using a two-tailed Student's t test with a threshold for rejecting the null hypothesis at the 95 % confidence level (see Sect. ). The robustness of the CCMI-1 multi-model mean SOR is assessed by distinguishing regions where the magnitude of the SOR is greater than ±2 SD of the intermodel spread. Figure  can be compared with Fig. 1 in and Fig. 1 in , which show similar plots for the CCMVal-1 and CMIP5 models, respectively.

All of the CCMI-1 models analysed show a significant positive SOR of up to ∼ 2 % between 1 and 10 hPa. This is less than half the peak amplitude of the SOR in the SAGE II v6.2 mixing ratio dataset and is more comparable to the SOR amplitude in SAGE II v7.0 mixing ratios and the SBUVMOD VN8.6 dataset (see Figs. 4 and 12 in ). The results from the CCMI-1 models are broadly consistent with the results from CCMVal-1 models . The main exception is the absence in the multi-model mean of a strong SOR in the tropical lower stratosphere. An enhanced SOR in the tropical lower stratosphere has been identified in satellite observations, albeit with large uncertainties , and it has been postulated this may be associated with a change in the strength of the Brewer–Dobson circulation. The CCMVal-1 multi-model mean showed a SOR of around 5 % per 130 SFU at ∼ 50 hPa (see Fig. 4d in ), as compared to around 1 % in the CCMI-1 multi-model mean (Fig. i). However, there was large intermodel spread in this signal across the CCMVal-1 models, and the multi-model mean SOR was dominated by strong responses in a few models that only ran for a short period (1980–2004), during which aliasing effects with other climatic factors can be significant . Since the analysis shown here extends for a longer period and excludes the post-volcanic epochs, this is a plausible reason for the apparent difference in the SOR in the tropical lower stratosphere between the CCMI-1 and CCMVal-1 models. One of the CCMI-1 models (SOCOL3) appears to show an enhanced SOR in the tropical lower stratosphere, which is similar in amplitude to that seen in some CCMVal-1 models. However, this feature shows some sensitivity to the choice of autoregressive model in the MLR model probably because the decorrelation timescale for the regression residuals in the tropical lower stratosphere is longer than 2 months in SOCOL3 and some of the other CCMs (not shown). Further analysis of the transformed Eulerian mean residual vertical velocity does not reveal a substantial change in the rate of upwelling in the tropical lower stratosphere in any of the models (not shown).

The month-by-month SORs in the individual CCMI-1 models (see Figs. S3–S10) show a significant positive SOR in the tropical upper stratosphere throughout the year but enhanced SOR amplitudes at high latitudes, particularly in the winter and spring seasons. This behaviour, which is also seen in some satellite ozone datasets (e.g. ), cannot be understood from photochemical processes alone and must therefore be related to stratospheric circulation changes (e.g. ). Such localized changes in ozone will be associated with a radiative perturbation that could lead to feedbacks onto circulation , and thus it may be important to account for this seasonal variation in the SOR in model simulations.

Figure  shows time series from 1960 to 2011 of deseasonalized tropical and monthly mean percent ozone anomalies at select stratospheric levels (1 to 30 hPa) from the Bodeker (orange line), CMIP5 (red) and CMIP6 (blue) ozone databases. Also plotted in black are the same satellite datasets shown in Fig. . Anomalies are defined relative to the period 1985–2007. The Extended CMIP5 ozone database is not shown because it is identical to the original CMIP5 database in the tropics.

Although the time series have been deseasonalized, the CMIP5 and Bodeker ozone databases show a residual annual cycle particularly in the upper stratosphere. This is because in these databases the amplitude of the ozone annual cycle is larger in the early part of the record, when the background levels are higher, and smaller in the latter part of the record following the long-term decline in ozone. Since the ozone anomalies in Fig.  are shown as anomalies from the 1985–2007 mean, there is therefore a residual annual cycle particularly in the period before 1985. Conversely, the CMIP6 database, which is constructed from CCM simulations, does not show a significant change in the amplitude of the ozone annual cycle over time.

At 1 hPa, the CMIP5 and Bodeker databases show a larger linear trend in ozone over 1979–2007 (diagnosed using linear regression) of around -3.5 %decade-1 compared to -1 %decade-1 in the CMIP6 database. The latter is, as expected, similar to the long-term ozone trends in the CCMI-1 models shown in Fig. . At 3 hPa, the CMIP5 database also shows a larger long-term decrease in ozone by around a factor of 2 compared to the Bodeker and CMIP6 databases. Thus, a model that uses the recommended CMIP6 ozone database might be expected to show weaker upper-stratospheric cooling over recent decades than an equivalent simulation using the CMIP5 database, owing to the smaller negative trend in upper-stratospheric ozone.

At 10 and 30 hPa, the Bodeker and CMIP6 databases show a QBO signal in ozone, whereas the CMIP5 database does not include QBO variability. This is an important distinction because a model that employs the CMIP6 ozone database, but which does not simulate a dynamical QBO, will impose a QBO-ozone signal that may alter the model's behaviour. Alternatively, a model that internally generates a dynamical QBO that is not in phase with the prescribed QBO-ozone signal in the CMIP6 ozone database will be subject to a diabatic heating anomaly from ozone that is inconsistent with the model's dynamical evolution. Both of these cases would be physically unrealistic. However, a model that nudges a QBO towards observations and uses the CMIP6 ozone database should have a more consistent representation of temporal variability in winds and ozone associated with the QBO. Modelling groups may therefore choose to post-process the CMIP6 ozone database in order to treat the QBO-ozone signal in a consistent manner for their model. Note that since the CMIP6 ozone database is produced by averaging two CCMs, one that does include QBO-ozone variability (CESM1(WACCM)) and one that does not (CMAM), the QBO-ozone signal is weaker in the CMIP6 ozone database than in the CESM1(WACCM) model alone (compare blue line in Fig.  with dark pink line in Fig. ). The absence of a QBO-ozone signal in the CMIP5 ozone database means that CMIP5 models that simulated a QBO would have neglected any radiative feedback from ozone on the QBO.

Figure  shows latitude–pressure cross sections of the annual mean SOR in the three ozone databases shown in Fig.  and the Extended CMIP5 ozone database. In the tropics, the Bodeker ozone database, Fig. a, shows a positive SOR of up to 4 % peaking at around 2–3 hPa with a distinct minimum at ∼ 10 hPa. The latitudinal structure of the SOR is smoother than in the SAGE II v6.2 mixing ratio data (cf. Fig. 4d of Part 1) probably because the construction of the Bodeker database fits an MLR model to all data points along pressure surfaces rather than to individual latitude bands. At high latitudes, the magnitude of the SOR in the Bodeker database is small and the spatial structure is noisy likely because of the small number of observations there. In the lower stratosphere, the Bodeker database indicates a positive SOR at most latitudes, as was found in a number of satellite ozone datasets in Part 1. However, the uncertainty in the magnitude of the SOR at these levels is comparatively large (see below).

The SOR in the CMIP5 ozone database, Fig. b, shows a very similar structure to that found in SAGE v6.2 mixing ratios (cf. Fig. 4d in Part 1), consistent with those data forming the backbone for the historical portion of the dataset . Note that the MLR fits were applied separately at each latitude band in the construction of the CMIP5 database, and this likely explains why the horizontal structure of the SOR is more heterogeneous than in the Bodeker ozone database. In particular the three-peaked structure of the SOR found in the tropical upper stratosphere in the SAGE II v6.2 mixing ratio dataset is evident in the CMIP5 ozone database. The sharp cut-offs in the SOR at ±60∘ latitude are spurious and result from a lack of data points to define a SOR at high latitudes. As described in Sect. , the Extended CMIP5 ozone database, Fig. c, applied an extrapolation to introduce a SOR in the extratropics. The details of this structure, which shows a positive SOR extending into the northern extratropics and a negative SOR at pressures greater than ∼ 5 hPa in the Southern Hemisphere poleward of 60∘ S, is likely to be subject to considerable uncertainties owing to the simple spatial filling method employed.

In the CMIP6 ozone database, Fig. d, the amplitude of the SOR is around 1–2 % in the upper stratosphere, which is, as expected, broadly consistent with the CCMI-1 results shown in Fig. . The peak amplitude of the SOR is therefore 2–3 times smaller, and is considerably smoother in latitude, than in the CMIP5 ozone database. In the lowermost tropical stratosphere, the CMIP6 database shows a positive SOR of up to ∼ 3 % in the southern tropics. This is slightly larger than the SOR in the tropical lower stratosphere simulated by the CCMI-1 versions of the two CCMs used to produce the CMIP6 ozone database (CESM1(WACCM) and CMAM) (see Fig. b and c). To further investigate the vertical structure of the tropical SOR and its uncertainties, Fig.  shows the best-estimate tropical (30∘ S–30∘ N) mean SOR along with the 2.5–97.5 % confidence intervals for the climate model ozone databases and the two satellite datasets from Fig.  (see Part 1). Also shown in grey shading is the range of the best-estimate SORs from the eight CCMI-1 models. The best-estimate SOR in the tropical lower stratosphere ranges from a small negative signal in the CMIP5 ozone database to +6 % in the Bodeker ozone database. In the CMIP6 ozone database, the best-estimate tropical SOR is 2 % at 80 hPa, which is, as expected, within the range of the signals in the CCMI-1 models. The substantial spread amongst the estimates along with the large uncertainties reinforces the challenge of constraining the SOR in the tropical lower stratosphere (e.g. ). Despite the relatively large uncertainties, the SOR in the tropical lower stratosphere is larger in the CMIP6 database than in CMIP5; this may be important for the modelled atmospheric response to solar variability in CMIP5 and CMIP6 models (see Sect. ). Figure  further confirms that the two climate model ozone databases that include SAGE II v6.2 mixing ratio data (CMIP5 and Bodeker) show a significantly stronger SOR in the tropical upper stratosphere. This is likely to be unrealistically large since the updated SAGE II v7.0 mixing ratio data, which show a smaller SOR in the tropical upper stratosphere , exhibit a more realistic representation of the relationship between upper-stratospheric ozone and temperature than the v6.2 data .

Earlier studies have shown evidence for an annual cycle in the structure and amplitude of the SOR in satellite observations (e.g. ). Figure  shows the monthly mean SOR in the Extended CMIP5 ozone database, and Fig.  shows the same for the CMIP6 ozone database. The SOR in the CMIP5 database has a fixed structure and constant amplitude in all months; the small annual cycle in the fractional SOR amplitude arises purely from the annual cycle in background ozone concentrations. There are well-understood photochemical arguments for why the structure of the SOR is expected to track the position of the Sun through the year . Furthermore, the coupling between ozone and stratospheric dynamics may lead to variations in the SOR at high latitudes in some months due to the formation in winter of the polar vortices and their subsequent break-up in spring . For these reasons a complete absence of seasonal variation in the SOR as found in the CMIP5 ozone database is unrealistic. In contrast, the SOR in the CMIP6 ozone database, Fig. , shows greater seasonal variation. Locally enhanced signals in the SOR are found in the high latitudes in winter and spring, which may be linked to variations in the strength of the polar vortex . Thus, including some semblance of an annual cycle in the SOR, as seen in Fig. , is likely to be a truer reflection of the behaviour of the real atmosphere than the complete absence of an SOR annual cycle as in Fig. . However, the associated uncertainties in the monthly SORs are larger than the annual mean results presented in the previous section, and there are quantitative differences between the SOR annual cycle in the CMIP6 ozone database and that estimated from satellite observations (cf. e.g. Fig. 13 of ). Such differences may result from uncertainties in estimating the SOR from relatively short observational records, from errors in the representation of the SOR in the models used to construct the CMIP6 ozone database or a combination of factors. Thus we should not consider the evolution of the monthly SOR in the CMIP6 ozone database as a precise representation of the true SOR, but it is likely an improvement compared to the representation in the CMIP5 ozone database.

We now explore the atmospheric impacts of the differences between the SOR in the CMIP5 and CMIP6 ozone databases using the ECHAM6.3 model sensitivity experiments described in Sect. . Figure  shows the tropical average annual mean temperature differences in the four perturbation experiments representing 11-year solar cycle maximum conditions with respect to the control solar minimum simulation. Note that the tropospheric temperature responses in all simulations are small because the model includes fixed SSTs, and therefore the troposphere does not fully adjust to the imposed solar forcing (e.g. ).

The experiments performed to capture the total (i.e. SSI + SOR) solar cycle impact (dashed lines) show considerable differences in the tropical mean stratospheric temperature response between the recommended CMIP5 (red line) and CMIP6 (blue line) solar forcings. In the CMIP5 case, the maximum temperature response is around 1.25 K near the stratopause, which can be compared to a much smaller response to the CMIP6 solar forcing inputs of 0.8 K. The SOR-only sensitivity experiments (solid lines) reveal that much of the difference in the total temperature response can be attributed to the differences in the SOR between the CMIP5 and CMIP6 ozone databases. The SOR in the Extended CMIP5 ozone database induces a peak tropical temperature response of 0.85 K (solid red), which is nearly 3 times larger than the maximum response to the SOR in the CMIP6 ozone database with an amplitude of 0.3 K (solid blue). In addition to the marked differences in the maximum temperature response, there are also distinct differences in vertical structure. In the CMIP5 case, there is a stronger vertical gradient in the temperature response to the imposed solar forcing, which can be attributed to the highly peaked structure of the SOR in the CMIP5 ozone database at the stratopause compared to the smoother vertical structure of the SOR in the CMIP6 database (cf. Fig. c and d). The simulation forced with the SOR from the CMIP6 ozone database also shows a small secondary peak in tropical lower-stratospheric temperature of ∼ 0.3 K due to the presence of a locally enhanced SOR of ∼ 3 %, which is not present in the CMIP5 ozone database. The results show that the change in the representation of the SOR between the recommended CMIP5 and CMIP6 ozone databases induces a much larger difference in the stratospheric temperature response between solar cycle minimum and maximum than do changes in the recommended SSI forcing (see also Fig. 8 in ).

The ECHAM6.3 model results help to elucidate the findings of , which show a clear difference in the annual mean stratospheric temperature response to the solar cycle between CMIP5 models that used the CMIP5 ozone database (HadGEM2-CC, MPI-ESM, CMCC) and those with interactive chemistry that simulated their own internally consistent SOR (CESM1(WACCM), GFDL-CM3, GISS-E2-H, MIROC-ESM-CHEM, MRI-ESM1). Specifically, models that used the CMIP5 ozone database exhibit a markedly larger temperature response near the tropical stratopause, with a stronger vertical gradient, than the models with interactive chemistry (see Fig. 5 in ). One might therefore anticipate that the difference in the stratospheric temperature response to the solar cycle between models with and without interactive chemistry will be smaller in CMIP6 than was found in CMIP5 owing to the fact that the SOR in the CMIP6 ozone database is derived from CCM simulations, albeit without full consistency with the other CMIP6 external forcings such as SSI.